CONVERGENCE OF GENERIC INFINITE
PRODUCTS OF AFFINE OPERATORS
SIMEON REICH AND ALEXANDER J. ZASLAVSKI
Received 23 November 1998
We establish several results concerning the asymptotic behavior of random infinite
products of generic sequences of affine uniformly continuous operators on bounded
closed convex subsets of a Banach space. In addition to weak ergodic theorems we also
obtain convergence to a unique common fixed point and more generally, to an affine
retraction.
1. Introduction
Our goal in this paper is to study the asymptotic behavior of random infinite products of
generic sequences of affine uniformly continuous operators on bounded closed convex
subsets of a Banach space. Infinite products of operators find application in many areas
of mathematics (cf. [1, 2, 3, 8, 9, 10] and the references therein). More precisely, we
show that in appropriate spaces of sequences of operators there exists a subset which
is a countable intersection of open everywhere dense sets such that for each sequence
belonging to this subset the corresponding random infinite products converge. Results
of this kind for powers of a single nonexpansive operator were already established
by De Blasi and Myjak [6] while such results for infinite products have recently been
obtained in [13]. The approach used in these papers and in the present paper is common
in global analysis and in the theory of dynamical systems [7, 11]. Recently it has also
been used in the study of the structure of extremals of variational and optimal control
problems [14, 15, 16]. Thus, instead of considering a certain convergence property for
a single sequence of affine operators, we investigate it for a space of all such sequences
equipped with some natural metric, and show that this property holds for most of these
sequences. This allows us to establish convergence without restrictive assumptions on
the space and on the operators themselves. We remark in passing that common fixed
point theorems for families of affine mappings (e.g., those of Markov-Kakutani and
Ryll-Nardzewski) have applications in various mathematical areas. See, for example,
[5] and the references therein.
Copyright © 1999 Hindawi Publishing Corporation
Abstract and Applied Analysis 4:1 (1999) 1–19
1991 Mathematics Subject Classification: 47H10, 58F99, 54E52
URL: http://aaa.hindawi.com/volume-4/S1085337599000032.html
2
Convergence of generic infinite products of affine operators
Let (X, · ) be a Banach space and let K be a nonempty bounded closed convex
subset of X with the topology induced by the norm · .
Denote by A the set of all sequences {At }∞
t=1 , where each At : K → K is a continuous
operator, t = 1, 2, . . . . Such a sequence will occasionally be denoted by a boldface A.
We equip the set A with the metric ρs : A × A → [0, ∞) defined by
∞
ρs {At }∞
t=1 , {Bt }t=1 = sup At x − Bt x : x ∈ K, t = 1, 2, . . . ,
∞
{At }∞
t=1 , {Bt }t=1 ∈ A.
(1.1)
It is easy to see that the metric space (A, ρs ) is complete. We always consider the set
A with the topology generated by the metric ρs .
We say that a set E of operators A : K → K is uniformly equicontinuous (ue) if for
any > 0 there exists δ > 0 such that Ax − Ay ≤ for all A ∈ E and all x, y ∈ K
satisfying x − y ≤ δ.
An operator A : K → K is called uniformly continuous if the singleton {A} is a
(ue) set.
Define
∞
Aue = {At }∞
(1.2)
t=1 ∈ A : {At }t=1 is a (ue) set .
Clearly Aue is a closed subset of A.
We endow the topological subspace Aue ⊂ A with the relative topology.
We say that an operator A : K → K is affine if
A αx + (1 − α)y = αAx + (1 − α)Ay
(1.3)
for each x, y ∈ K and all α ∈ [0, 1].
Denote by M the set of all uniformly continuous affine mappings A : K → K. For
the space M we consider the metric
ρ(A, B) = sup{Ax − Bx : x ∈ K},
A, B ∈ M.
(1.4)
It is easy to see that the metric space (M, ρ) is complete.
In the present paper, we analyze the convergence of infinite products of operators in
M and other mappings of affine type.
We begin by showing (Theorem 3.1) that for a generic operator B in the space M
there exists a unique fixed point xB and the powers of B converge to xB for all x ∈ K.
We continue with a study of the asymptotic behavior of infinite products of this kind
of operators. Section 2 contains necessary preliminaries and a weak ergodic theorem
is established in Section 4. In Sections 5 and 7 we present several theorems on the
generic convergence of infinite product trajectories to a common fixed point and to a
common fixed point set, respectively. Proofs of these results are given in Sections 6
and 8. Finally, in Section 9 we establish the generic convergence of random products
to a retraction onto a common fixed point set.
S. Reich and A. J. Zaslavski
3
2. Infinite products
∞
Denote by Aaf
ue the set of all {At }t=1 ∈ Aue such that for each integer t ≥ 1, each
x, y ∈ K and all α ∈ [0, 1],
(2.1)
At αx + (1 − α)y = αAt x + (1 − α)At y.
af
Clearly Aaf
ue is a closed subset of Aue . We consider the topological subspace Aue ⊂ Aue
with the relative topology.
In this paper we show (Theorem 4.1) that for a generic sequence {Ct }∞
t=1 in the space
af
Aue ,
Cr(T ) · · · · · Cr(1) x − Cr(T ) · · · · · Cr(1) y → 0
(2.2)
uniformly for all x, y ∈ K and all mappings r : {1, 2, . . .} → {1, 2, . . .}. Such results are
usually called weak ergodic theorems in the population biology literature [4] (see also
[12]).
Denote by A0ue the set of all A = {At }∞
t=1 ∈ Aue for which there exists xA ∈ K
such that
At xA = xA , t = 1, 2, . . . ,
(2.3)
and for each γ ∈ (0, 1), x ∈ K and each integer t ≥ 1,
At γ xA + (1 − γ )x = λt (γ , x)xA + 1 − λt (γ , x) At x
(2.4)
with some constant λt (γ , x) ∈ [γ , 1].
Denote by Ā0ue the closure of A0ue in the space Aue . We consider the topological
subspace Ā0ue with the relative topology and show (Theorem 5.1) that for a generic
0
sequence {Ct }∞
t=1 in the space Āue there exists a unique common fixed point x∗ and all
random products of the operators {Ct }∞
t=1 converge to x∗ uniformly for all x ∈ K. We
also show that this convergence of random infinite products to a unique common fixed
point holds for a generic sequence from certain subspaces of the space Ā0ue .
Assume now that F ⊂ K is a nonempty closed convex set, Q : K → F is a uniformly
continuous operator such that
Qx = x,
x ∈ F,
and for each y ∈ K, x ∈ F and α ∈ [0, 1],
Q αx + (1 − α)y = αx + (1 − α)Qy.
(F,0)
Denote by Aue
t = 1, 2, . . . , x ∈ F,
and for each integer t ≥ 1, each y ∈ K, x ∈ F and α ∈ (0, 1],
At αx + (1 − α)y = αx + (1 − α)At y.
(F,0)
(2.6)
the set of all {At }∞
t=1 ∈ Aue such that
At x = x,
Clearly Aue
(2.5)
is a closed subset of Aue .
(2.7)
(2.8)
4
Convergence of generic infinite products of affine operators
(F,0)
The topological subspace Aue ⊂ Aue will be equipped with the relative topology.
We show (Theorem 7.1) that for a generic sequence of operators {Ct }∞
t=1 in the space
(F,0)
Aue all its random infinite products
Cr(t) · · · · · Cr(1) x
(2.9)
tend to the set F uniformly for all x ∈ K. Moreover, under a certain additional assumption on F these random products converge to a uniformly continuous retraction
Pr : K → F uniformly for all x ∈ K (Theorem 9.1).
For each bounded operator A : K → X we set
A = sup{Ax : x ∈ K}.
(2.10)
For each x ∈ K and each E ⊂ X we set
d(x, E) = inf{x − y : y ∈ E},
rad(E) = sup{y : y ∈ E}.
(2.11)
In our study we need the following auxiliary result established in [13, Lemma 4.2].
Proposition 2.1. Assume that E is a nonempty uniformly continuous set of operators
A : K → K, N is a natural number and is a positive number. Then there exists a
N
number δ > 0 such that for each sequence {At }N
t=1 ⊂ E, each sequence {Bt }t=1 , where
the operators Bt : K → K, t = 1, . . . , N , (not necessarily continuous), satisfy
Bt − At ≤ δ, t = 1, . . . , N,
(2.12)
and each x ∈ K, the following relation holds:
BN · · · · · B1 x − AN · · · · · A1 x ≤ .
(2.13)
3. Existence of a unique fixed point for a generic affine mapping
This section is devoted to the proof of the following result.
Theorem 3.1. There exists a set F ⊂ M which is a countable intersection of open
everywhere dense subsets of M such that for each A ∈ F the following assertions hold:
(1) there exists a unique xA ∈ K such that AxA = xA ;
(2) for each > 0 there exist a neighborhood U of A in M and a natural number
N such that for each {Bt }∞
t=1 ⊂ U and each x ∈ K,
BT · · · · · B1 x − xA ≤ for all integers T ≥ N.
(3.1)
In the proof of Theorem 3.1 we need the following lemma.
Lemma 3.2. Let B ∈ M and ∈ (0, 1). Then there exist B ∈ M, an integer q ≥ 1, and
y ∈ K such that
ρ(B, B ) ≤ , Bt y − y ≤ , t = 1, . . . , q,
(3.2)
S. Reich and A. J. Zaslavski
5
and for each z ∈ K the following relation holds:
q
B z − y ≤ .
(3.3)
Proof. Choose a number γ ∈ (0, 1) for which
8γ rad(K) + 1 ≤ ,
(3.4)
and then an integer q ≥ 1 such that
(1 − γ )q rad(K) + 1 ≤ 16−1 ,
(3.5)
and a natural number N such that
16qN −1 rad(K) + 1 ≤ 8−1 .
(3.6)
Fix x0 ∈ K and define a sequence {xt }∞
t=0 ⊂ K by
xt+1 = Bxt ,
t = 0, 1, . . . .
(3.7)
For each integer k ≥ 0 define
yk = N −1
k+N−1
xi .
(3.8)
i=k
It is easy to see that
Byk = yk+1 ,
k = 0, 1, . . .
(3.9)
and for each k ∈ {0, . . . , q}
y0 − yk ≤ 2kN −1 rad(K) ≤ 2qN −1 rad(K).
(3.10)
Define B : K → K by
B z = (1 − γ )Bz + γ y0 ,
z ∈ K.
(3.11)
It is easy to see that
B ∈ M
and
ρ(B, B ) < 2−1 .
(3.12)
Now let z be an arbitrary point in K. We show by induction that for each integer n ≥ 1
Bn z = (1 − γ )n B n z +
where
cni > 0,
i = 0, . . . , n − 1,
n−1
cni yi ,
(3.13)
i=0
n−1
i=0
It is easy to see that for n = 1 our assertion holds.
cni + (1 − γ )n = 1.
(3.14)
6
Convergence of generic infinite products of affine operators
Assume that it is also valid for an integer n ≥ 1. It follows from (3.11), (3.13), (3.14),
(3.12), and (3.9) that
Bn+1 z = γ y0 + (1 − γ )B Bn z
n−1
n n+1
z+
cni Byi
= γ y0 + (1 − γ ) (1 − γ ) B
(3.15)
i=0
= (1 − γ )n+1 B n+1 z + γ y0 + (1 − γ )
n−1
cni yi+1 .
i=0
This implies that our assertion is also valid for n + 1. Therefore for each integer n ≥ 1,
(3.13) holds with some constants cni , i = 0, . . . , n − 1, satisfying (3.14).
Now we show that
q
B z − y0 ≤ .
(3.16)
We have shown that there exist positive numbers cqi > 0, i = 0, . . . , q − 1, such that
q−1
q
cqi + (1 − γ ) = 1
Bq z
and
i=0
q
q
= (1 − γ ) B z +
q−1
cqi yi .
(3.17)
i=0
By (3.17), (3.10), (3.5), and (3.6),
q−1
q
B z − y0 ≤ (1 − γ )q B q z − y0 +
cqi y0 − yi i=0
−1
q
≤ 2(1 − γ ) rad(K) + 2qN
(3.18)
rad(K)
≤ 16−1 + 8−1 < 2−1 .
Therefore we have shown that
q
B z − y0 ≤ 2−1 for each z ∈ K.
(3.19)
Let t ∈ {1, . . . , q}. To finish the proof we show that
t
B y0 − y0 ≤ .
(3.20)
By (3.13) and (3.14) there exist positive numbers cti , i = 0, . . . , t − 1, such that
t−1
i=0
cti + (1 − γ )t = 1
and
Bt y0 = (1 − γ )t B t y0 +
t−1
≤ 4qN
(3.21)
i=0
Together with (3.9), (3.10), and (3.6) this implies that
t−1
t y0 − B t y0 = y0 −
cti yi − (1 − γ ) yt i=0
−1
cti yi .
(3.22)
rad(K) < 8−1 .
This completes the proof of Lemma 3.2 (with y = y0 ).
S. Reich and A. J. Zaslavski
7
Proof of Theorem 3.1. To begin the construction of the set F, let B ∈ M and let i ≥ 1
be an integer. By Lemma 3.2 there exist C (B,i) ∈ M, y(B, i) ∈ K, and an integer
q(B, i) ≥ 1 such that
ρ B, C (B,i) ≤ 8−i ,
(B,i) t
C
y(B, i) − y(B, i) ≤ 8−i ,
(B,i) q(B,i)
C
z − y(B, i) ≤ 8−i
t = 0, . . . , q(B, i),
(3.23)
for each z ∈ K.
(3.24)
By Proposition 2.1 there exists an open neighborhood U (B, i) of C (B,i) in M such that
q(B,i)
for each {Aj }j =1 ⊂ U (B, i) and each z ∈ K,
Aq(B,i) · · · · · A1 z − C (B,i) q(B,i) z ≤ 64−i .
(3.25)
q(B,i)
It follows from (3.24) and (3.25) that for each {Ai }j =1 ⊂ U (B, i) and each z ∈ K,
Aq(B,i) · · · · · A1 z − y(B, i) ≤ 8−i + 64−i .
(3.26)
F = ∩∞
k=1 ∪ {U (B, i) : B ∈ M, i = k, k + 1, . . .}.
(3.27)
Define
It is easy to see that F is a countable intersection of open everywhere dense subsets
of M.
Assume that A ∈ F and > 0. Choose a natural number k for which
64 · 2−k < .
(3.28)
There exist B ∈ M and an integer i ≥ k such that
A ∈ U (B, i).
Combined with (3.26) and (3.28) this implies that for each z ∈ K,
q(B,i)
A
z − y(B, i) ≤ 8−i + 64−i < .
(3.29)
(3.30)
Since is an arbitrary positive number we conclude that there exists a unique xA ∈ K
such that AxA = xA . Clearly
xA − y(B, i) ≤ 8−i + 64−i .
(3.31)
Together with (3.26) and (3.28) this last inequality implies that for each {Aj }∞
j =1 ⊂
U (B, i), each z ∈ K, and each integer T ≥ q(B, i),
AT · · · · · A1 z − xA ≤ 2 8−i + 64−i < .
(3.32)
This completes the proof of Theorem 3.1.
8
Convergence of generic infinite products of affine operators
4. A weak ergodic theorem for infinite products of affine mappings
In this section we establish the following result.
Theorem 4.1. There exists a set F ⊂ Aaf
ue which is a countable intersection of open
everywhere dense subsets of Aaf
such
that
for each {Bt }∞
ue
t=1 ∈ F and each > 0 there
∞
af
exists a neighborhood U of {Bt }t=1 in Aue and a natural number N such that for each
{Ct }∞
t=1 ∈ U , each integer T ≥ N , each r : {1, . . . , T } → {1, 2, . . .}, and each x, y ∈ K,
Cr(T ) · · · · · Cr(1) x − Cr(T ) · · · · · Cr(1) y ≤ .
(4.1)
af
Proof. Fix y∗ ∈ K. Let {At }∞
t=1 ∈ Aue and γ ∈ (0, 1). For t = 1, 2, . . . define Atγ :
K → K by
Atγ x = (1 − γ )At x + γ y∗ , x ∈ K.
(4.2)
Clearly
Atγ
∞
t=1
∈ Aaf
ue ,
∞ ∞ ρs At t=1 , Atγ t=1 ≤ 2γ rad(K).
Let i ≥ 1 be an integer. Choose a natural number N (γ , i) ≥ 4 such that
(1 − γ )N (γ ,i) rad(K) + 1 < 16−1 4−i .
(4.3)
(4.4)
We show by induction that for each integer T ≥ 1 the following assertion holds.
For each r : {1, . . . , T } → {1, 2, . . .} there exists yr,T ∈ K such that
Ar(T )γ · · · · · Ar(1)γ x = (1 − γ )T Ar(T ) · · · · · Ar(1) x + 1 − (1 − γ )T yr,T
(4.5)
for each x ∈ K.
Clearly for T = 1 the assertion is true. Assume that it is also true for an integer
T ≥ 1. It follows from (4.5) that for each r : {1, . . . , T + 1} → {1, 2, . . .} and each
x ∈ K,
Ar(T +1)γ · · · · · Ar(1)γ x
= Ar(T +1)γ Ar(T )γ · · · · · Ar(1)γ x
= Ar(T +1)γ (1 − γ )T Ar(T ) · · · · · Ar(1) x + 1 − (1 − γ )T yr,T
= γ y∗ + (1 − γ )Ar(T +1) (1 − γ )T Ar(T ) · · · · · Ar(1) x + 1 − (1 − γ )T yr,T
= (1 − γ )T +1 Ar(T +1) · · · · · Ar(1) x + (1 − γ ) 1 − (1 − γ )T Ar(T +1) yr,T + γ y∗ .
(4.6)
This implies that the assertion is also valid for T + 1. Therefore, we have shown
that our assertion is true for any integer T ≥ 1. Together with (4.4) this implies that the
following property holds:
(a) for each integer T ≥ N (γ , i), each r : {1, . . . , T } → {1, 2, . . .}, and each x, y ∈ K,
Ar(T )γ · · · · · Ar(1)γ x − Ar(T )γ · · · · · Ar(1)γ y ≤ 2(1 − γ )T rad(K) ≤ 8−1 · 4−i . (4.7)
S. Reich and A. J. Zaslavski
9
∞
By Proposition 2.1 there is an open neighborhood U ({At }∞
t=1 , γ , i) of {Atγ }t=1
∞
∞
af
in Aue such that for each {Ct }t=1 ∈ U ({At }t=1 , γ , i), each r : {1, . . . , N (γ , i)} →
{1, 2, . . .}, and each x ∈ K,
Cr(N(γ ,i)) · · · · · Cr(1) x − Ar(N (γ ,i))γ · · · · · Ar(1)γ x ≤ 64−1 · 4−i .
(4.8)
Together with property (a) this implies that the following property holds:
(b) for each integer T ≥ N (γ , i), each r : {1, . . . , T } → {1, 2, . . .}, each x, y ∈ K,
∞
and each {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i),
Cr(T ) · · · · · Cr(1) x − Cr(T ) · · · · · Cr(1) y ≤ 4−i−1 .
(4.9)
Define
∞
∞
af
F = ∩∞
q=1 ∪ U {At }t=1 , γ , i : {At }t=1 ∈ Aue , γ ∈ (0, 1), i = q, q + 1, . . . . (4.10)
Clearly F is a countable intersection of open everywhere dense subsets of Aaf
ue . Let
{Bt }∞
∈
F
and
>
0.
Choose
a
natural
number
q
for
which
t=1
64 · 2−q < .
af
There exist {At }∞
t=1 ∈ Aue , γ ∈ (0, 1), and an integer i ≥ q such that
∞
{Bt }∞
t=1 ∈ U {At }t=1 , γ , i .
(4.11)
(4.12)
∞
By property (b) and (4.11), for each {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i), each T ≥ N (γ , i), each
r : {1, . . . , T } → {1, 2, . . .}, and each x, y ∈ K,
Cr(T ) · · · · · Cr(1) x − Cr(T ) · · · · · Cr(1) y ≤ 4−i−1 < .
(4.13)
This completes the proof of Theorem 4.1.
5. The convergence of infinite products of affine mappings
with a common fixed point
In this section we state three theorems which will be proved in Section 6.
Theorem 5.1. There exists a set F ⊂ Ā0ue which is a countable intersection of open
everywhere dense subsets of Ā0ue such that F ⊂ A0ue and for each B = {Bt }∞
t=1 ∈ F the
following assertion holds.
Let xB ∈ K, Bt xB = xB , t = 1, 2, . . . , and let > 0. Then there exist a neighborhood
∞
0
U of B = {Bt }∞
t=1 in Āue and a natural number N such that for each {Ct }t=1 ∈ U , each
integer T ≥ N, each r : {1, . . . , T } → {1, 2, . . .}, and each x ∈ K,
Cr(T ) · · · · · Cr(1) x − xB ≤ .
(5.1)
(1)
Denote by Aue the set of all A = {At }∞
t=1 ∈ Aue for which there exists xA ∈ K
such that
At xA = xA , t = 1, 2, . . . ,
(5.2)
10
Convergence of generic infinite products of affine operators
and for each α ∈ (0, 1), x ∈ K, and an integer t ≥ 1,
At αxA + (1 − α)x = αxA + (1 − α)At x.
(1)
(5.3)
(1)
Denote by Āue the closure of Aue in the space Aue . We equip the topological
(1)
subspace Āue ⊂ Aue with the relative topology.
Theorem 5.2. Let a set F ⊂ Ā0ue be as guaranteed in Theorem 5.1. There exists a set
(1)
F(1) ⊂ F ∩ Aue which is a countable intersection of open everywhere dense subsets of
(1)
Āue .
∞
af
Denote by Aaf
ue,0 the set of all A = {At }t=1 ∈ Aue for which there exists xA ∈ K
such that (5.2) holds.
af
Denote by Āaf
ue,0 the closure of Aue,0 in the space Aue . We also consider the topological subspace Āaf
ue,0 ⊂ Aue with the relative topology.
Theorem 5.3. Let a set F(1) be as guaranteed in Theorem 5.2. There exists a set
F∗ ⊂ F(1) ∩ Aaf
ue,0 which is a countable intersection of open everywhere dense subsets
of Āaf
.
ue,0
Theorems 5.2 and 5.3 show that the generic convergence established in Theorem 5.1
is also valid for certain subspaces of Ā0ue .
6. Proofs of Theorems 5.1, 5.2, and 5.3
0
Proof of Theorem 5.1. Let A = {At }∞
t=1 ∈ Aue and γ ∈ (0, 1). There exists xA ∈ K
such that
A t xA = x A ,
t = 1, 2, . . . ,
(6.1)
and for each integer t ≥ 1, x ∈ K, and α ∈ (0, 1),
At αxA + (1 − α)x = λt (α, x)xA + 1 − λt (α, x) At x
(6.2)
with some constant λt (α, x) ∈ [α, 1].
For t = 1, 2, . . . define Atγ : K → K by
Atγ x = (1 − γ )At x + γ xA ,
x ∈ K.
(6.3)
t = 1, 2, . . .
(6.4)
Clearly
Atγ
∞
t=1
∈ Aue ,
Atγ xA = xA ,
S. Reich and A. J. Zaslavski
11
Let x ∈ K, α ∈ [0, 1) and let t ≥ 1 be an integer. Then there exists λt (α, x) ∈ [α, 1]
such that (6.2) holds. Also, by (6.3) and (6.2),
Atγ αxA + (1 − α)x
= (1 − γ )At αxA + (1 − α)x + γ xA
= γ xA + (1 − γ ) λt (α, x)xA + 1 − λt (α, x) At x
(6.5)
= (1 − γ ) 1 − λt (α, x) At x + γ + (1 − γ )λt (α, x) xA
= 1 − λt (α, x) Atγ x + γ + (1 − γ )λt (α, x) − γ 1 − λt (α, x) xA
= 1 − λt (α, x) Atγ x + λt (α, x)xA .
Thus property (2.4) is satisfied and therefore
∞
Atγ t=1 ∈ A0ue .
(6.6)
Let z ∈ K. We show by induction that for each integer T ≥ 1 and each r : {1, . . . , T } →
{1, 2, . . .} there exists λ(z, T , r) ∈ [0, (1 − γ )T ] such that
(6.7)
Ar(T )γ · · · · · Ar(1)γ z = λ(z, T , r)Ar(T ) · · · · · Ar(1) z + 1 − λ(z, T , r) xA .
Clearly for T = 1 our assertion is valid.
Assume that it is also valid for an integer T ≥ 1. Let r : {1, . . . , T + 1} → {1, 2, . . .}.
There exists λ(z, T , r) ∈ [0, (1 − γ )T ] such that (6.7) is valid. It follows from (6.7) and
(6.5) that
Ar(T +1)γ · · · · · Ar(1)γ z
= Ar(T +1)γ λ(z, T , r)Ar(T ) · · · · · Ar(1) z + 1 − λ(z, T , r) xA
= (1 − γ )(1 − κ)Ar(T +1) Ar(T ) · · · · · Ar(1) z + γ + (1 − γ )κ xA
(6.8)
with κ ∈ [1 − λ(z, T , r), 1]. Set
λ(z, T + 1, r) = (1 − γ )(1 − κ).
(6.9)
It is easy to see that
0 ≤ λ(z, T + 1, r) ≤ (1 − γ )λ(z, T , r) ≤ (1 − γ )T +1 ,
Ar(T +1)γ · · · · · Ar(1)γ z
(6.10)
= λ(z, T + 1, r)Ar(T +1) · · · · · Ar(1) z + 1 − λ(z, T + 1, r) xA .
Therefore the assertion is valid for T + 1. We have shown that for each integer T ≥ 1
and each r : {1, . . . , T } → {1, 2, . . .} there exists λ(z, T , r) ∈ [0, (1 − γ )T ] such that
(6.7) holds.
Let i ≥ 1 be an integer. Choose a natural number N (γ , i) for which
64(1 − γ )N (γ ,i) rad(K) + 1 < 8−i .
(6.11)
12
Convergence of generic infinite products of affine operators
We show that for each z ∈ K, each integer T ≥ N (γ , i) and each r : {1, . . . , T } →
{1, 2, . . .},
Ar(T )γ · · · · · Ar(1)γ z − xA ≤ 8−i−1 .
(6.12)
Let T ≥ N (γ , i) be an integer, z ∈ K, and r : {1, . . . , T } → {1, 2, . . .}. There exists
λ(z, T , r) ∈ [0, (1 − γ )T ] such that (6.7) holds. It is easy to see that (6.7) and (6.11)
imply (6.12).
By Proposition 2.1 there exists a number
−1 −i
δ {At }∞
(6.13)
t=1 , γ , i ∈ 0, 16 8
0
such that for each {Ct }∞
t=1 ∈ Āue satisfying
∞
∞
ρs {Ct }∞
t=1 , {Atγ }t=1 ≤ δ {At }t=1 , γ , i ,
(6.14)
each r : {1, . . . , N(γ , i)} → {1, 2, . . .} and each x ∈ K,
Cr(N(γ ,i)) · · · · · Cr(1) x − Ar(N (γ ,i))γ · · · · · Ar(1)γ x ≤ 16−1 · 8−i .
(6.15)
Set
∞
0
∞
∞
∞
U {At }∞
t=1 , γ , i = {Ct }t=1 ∈ Āue : ρs {Ct }t=1 , {Atγ }t=1 < δ {At }t=1 , γ , i .
(6.16)
It follows from (6.16), the choice of δ({At }∞
,
γ
,
i)
(see
(6.13),
(6.15))
and
(6.12)
that
t=1
the following property holds:
∞
(a) for each {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i), each integer T ≥ N (γ , i), each r : {1, . . . , T }
→ {1, 2, . . .}, and each x ∈ K,
Cr(T ) · · · · · Cr(1) x − xA ≤ 8−i .
(6.17)
Define
∞
∞
0
F = ∩∞
q=1 ∪ U {At }t=1 , γ , i : {At }t=1 ∈ Aue , γ ∈ (0, 1), i = q, q + 1, . . . . (6.18)
It is easy to see that F is a countable intersection of open everywhere dense subsets of
Ā0ue .
Assume now that B = {Bt }∞
t=1 ∈ F and > 0. Choose a natural number q such that
64 · 2−q < .
0
There exist {At }∞
t=1 ∈ Aue , γ ∈ (0, 1), and an integer i ≥ q such that
∞
Bt t=1 ∈ U {At }∞
t=1 , γ , i .
(6.19)
(6.20)
By property (a), (6.20), and (6.19), for each x ∈ K, each integer T ≥ N (γ , i), and each
integer τ ≥ 1,
T
B x − xA ≤ 8−i < .
(6.21)
τ
Since is an arbitrary positive number we conclude that there exists xB ∈ K such that
lim BτT x = xB
T →∞
(6.22)
S. Reich and A. J. Zaslavski
for each x ∈ K and each integer τ ≥ 1. It is easy to see that
Bt xB = xB , t = 1, 2, . . . , xB − xA ≤ 8−i < .
13
(6.23)
It follows from property (a), (6.23), and (6.19) that for each sequence {Ct }∞
t=1 ∈
∞
U ({At }t=1 , γ , i), each integer T ≥ N (γ , i), each r : {1, . . . , T } → {1, 2, . . .}, and each
x ∈ K,
Cr(T ) · · · · · Cr(1) x − xB < .
(6.24)
We show that for each integer t ≥ 1, x ∈ K, and α ∈ (0, 1) there exists λ ∈ [α, 1]
such that
Bt αxB + (1 − α)x = λxB + (1 − λ)Bt x.
(6.25)
Let t ≥ 1 be an integer, x ∈ K and let α ∈ (0, 1). By (6.2) and (6.5) there exists
λ ∈ [α, 1] such that
Atγ αxA + (1 − α)x = λ xA + (1 − λ )Atγ x.
(6.26)
Since is an arbitrary positive number it follows from (6.26), (6.23), (6.20), (6.16),
(6.13), and (6.19) that for each > 0 there exist λ ∈ [α, 1], z ∈ K such that
z − xB ≤ , Bt αz + (1 − α)x − λ xB + (1 − λ )Bt x ≤ .
(6.27)
This implies that (6.25) holds with some λ ∈ [α, 1]. This completes the proof of
Theorem 5.1.
Proof of Theorem 5.2. Let F be as constructed in the proof of Theorem 5.1. Let A =
(1)
{At }∞
t=1 ∈ Aue , γ ∈ (0, 1) and let i ≥ 1 be an integer. There exists xA ∈ K such that
(6.15) holds, and for each x ∈ K, each integer t ≥ 1, and each α ∈ [0, 1] the equality
(6.2) holds with λt (α, x) = α. For t = 1, 2, . . . define Atγ : K → K by (6.3). It is easy
(1)
to see that {Atγ }∞
t=1 ∈ Aue . Choose a natural number N (γ , i) for which (6.11) holds.
∞
Let δ({At }t=1 , γ , i), U ({At }∞
t=1 , γ , i) be defined as in the proof of Theorem 5.1. Set
∞
F(1) = ∩∞
q=1 ∪ U {At }t=1 , γ , i
(6.28)
(1)
: {At }∞
∩ Ā(1)
ue .
t=1 ∈ Aue , γ ∈ (0, 1), i = q, q + 1, . . .
(1)
Clearly F(1) is a countable intersection of open everywhere dense subsets of Āue and
(1)
F(1) ⊂ F. Arguing as in the proof of Theorem 5.1 we can show that F(1) ⊂ Aue . This
completes the proof of Theorem 5.2.
Since the proof of Theorem 5.3 is analogous to that of Theorem 5.2 we omit it.
7. The weak convergence of infinite products of affine mappings
with a common set of fixed points
(F,0)
In this section, we present two theorems concerning the space Aue defined in Section
2. Recall that F is a nonempty closed convex subset of K for which there exists a
uniformly continuous operator Q : K → F such that
Qx = x,
x ∈ F,
(7.1)
14
Convergence of generic infinite products of affine operators
and for each y ∈ K, x ∈ F , and α ∈ [0, 1],
Q αx + (1 − α)y = αx + (1 − α)Qy.
(7.2)
Now we state the first theorem.
(F,0)
Theorem 7.1. There exists a set F ⊂ Aue which is a countable intersection of open
(F,0)
everywhere dense sets in Aue
and such that for each {Bt }∞
t=1 ∈ F the following
assertion holds.
(F,0)
For each > 0 there exist a neighborhood U of {Bt }∞
and
t=1 in the space Aue
∞
a natural number N such that for each {Ct }t=1 ∈ U , each integer T ≥ N, each r :
{1, 2, . . . , T } → {1, 2, . . .}, and each x ∈ K,
d Cr(T ) · · · · · Cr(1) x, F ≤ .
(7.3)
Assume now that for each x, y ∈ K and α ∈ [0, 1],
Q αx + (1 − α)y = αQx + (1 − α)Qy.
(F,1)
Denote by Aue
(7.4)
the set of all {At }∞
t=1 ∈ Aue such that
At x = x,
t = 1, 2, . . . , x ∈ F,
and for each t ∈ {1, 2, . . .}, each x, y ∈ K and each α ∈ [0, 1],
At αx + (1 − α)y = αAt x + (1 − α)At y.
(F,1)
(F,0)
Clearly Aue
is a closed subset of Aue
(F,1)
(F,0)
Aue ⊂ Aue with the relative topology.
Here is the second theorem.
(7.5)
(7.6)
. We consider the topological subspace
Theorem 7.2. Let a set F be as guaranteed in Theorem 7.1. Then there exists a set
(F,1)
F1 ⊂ F ∩ Aue which is a countable intersection of open everywhere dense subsets of
(F,1)
Aue .
8. Proof of Theorems 7.1 and 7.2
(F,0)
and γ ∈ (0, 1). For t = 1, 2, . . . we define
Proof of Theorem 7.1. Let {At }∞
t=1 ∈ Aue
Atγ : K → K by
Atγ x = (1 − γ )At x + γ Qx, x ∈ K.
(8.1)
It is easy to see that
(F,0)
{Atγ }∞
t=1 ∈ Aue .
(8.2)
Let z ∈ K. By induction we show that for each integer T ≥ 1 the following assertion
holds.
For each r : {1, . . . , T } → {1, 2, . . .},
Ar(T )γ · · · · · Ar(1)γ z = (1 − γ )T Ar(T ) · · · · · Ar(1) z + 1 − (1 − γ )T yT
(8.3)
with some yT ∈ F .
S. Reich and A. J. Zaslavski
15
Clearly for T = 1 our assertion is valid. Assume that it is also valid for T ≥ 1 and
that r : {1, . . . , T + 1} → {1, 2, . . .}. Clearly (8.3) holds with some yT ∈ F . By (8.3),
(8.2), and (8.1),
Ar(T +1)γ · · · · · Ar(1)γ z = Ar(T +1)γ (1 − γ )T Ar(T ) · · · · · Ar(1) z + 1 − (1 − γ )T yT
= (1 − γ )T Ar(T +1)γ Ar(T ) · · · · · Ar(1) z + 1 − (1 − γ )T yT
= (1 − γ )T +1 Ar(T +1) · · · · · Ar(1) z
+ γ (1 − γ )T Q[Ar(T ) · · · · · Ar(1) z] + 1 − (1 − γ )T yT
= (1 − γ )T +1 Ar(T +1) · · · · · Ar(1) z + 1 − (1 − γ )T +1
−1
× 1 − (1 − γ )T +1 γ (1 − γ )T Q Ar(T ) · · · · · Ar(1) z
−1 1 − (1 − γ )T ) yT .
+ 1 − (1 − γ )T +1
(8.4)
This implies that our assertion also holds for T + 1.
Therefore we have shown that it is valid for all integers T ≥ 1.
Let i ≥ 1 be an integer. Choose a natural number N (γ , i) for which
64(1 − γ )N (γ ,i) (rad(K) + 1) < 8−i .
(8.5)
It follows from (8.3) that for each z ∈ K, each integer T ≥ N (γ , i) and each r :
{1, . . . , T } → {1, 2, . . .},
d Ar(T )γ · · · · · Ar(1)γ z, F ≤ 8−i−1 .
(8.6)
∞
By Proposition 2.1 there exists an open neighborhood U ({At }∞
t=1 , γ , i) of {Atγ }t=1 in
(F,0)
Aue such that the following property holds:
∞
(a) for each {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i), each r : {1, . . . , N (γ , i)} → {1, 2, . . .}, and
each x ∈ K,
Cr(N(γ ,i)) · · · · · Cr(1) x − Ar(N (γ ,i))γ · · · · · Ar(1)γ x ≤ 16−1 8−i .
(8.7)
It follows from the definition of U ({At }∞
t=1 , γ , i) and (8.6) that the following property
is also true:
∞
(b) for each {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i), each integer T ≥ N (γ , i), each r : {1, . . . , T }
→ {1, 2, . . .} and each x ∈ K,
d Cr(T ) · · · · · Cr(1) x, F ≤ 8−i .
(8.8)
Define
∞
∞
(F,0)
F = ∩∞
q=1 ∪ U {At }t=1 , γ , i : {At }t=1 ∈ Aue , γ ∈ (0, 1), i = q, q + 1, . . . . (8.9)
It is easy to see that F is a countable intersection of open everywhere dense subsets of
(F,0)
Aue .
16
Convergence of generic infinite products of affine operators
Assume that {Bt }∞
t=1 ∈ F and > 0. Choose a natural number q such that
64 · 2−q < .
(8.10)
(F,0)
∞
There exist {At }∞
t=1 ∈ Aue , γ ∈ (0, 1), and an integer i ≥ q such that {Bt }t=1 ∈
∞
∞
∞
U ({At }t=1 , γ , i). By (8.10) and property (b) for each {Ct }t=1 ∈ U ({At }t=1 , γ , i), each
integer T ≥ N (γ , i), each r : {1, . . . , T } → {1, 2, . . .}, and each x ∈ K,
d Cr(T ) · · · · · Cr(1) x, F ≤ .
(8.11)
This completes the proof of Theorem 7.1.
Analogously to the proof of Theorem 5.2 we can prove Theorem 7.2 by modifying
the proof of Theorem 7.1.
9. The convergence of infinite products of affine mappings
with a common set of fixed points
In this section, as in Section 7, we assume that F is a nonempty closed convex subset
of K, and Q : K → F is a uniformly continuous retraction satisfying (7.2).
However we assume in addition that there exists a number ( > 0 such that
x ∈ X : d(x, F ) < ( ⊂ K.
(9.1)
In this setting we can strengthen Theorem 7.1.
(F,0)
Theorem 9.1. Let the set F ⊂ Aue be as constructed in the proof of Theorem 7.1.
Then for each {Bt }∞
t=1 ∈ F the following assertions hold:
(1) For each r : {1, 2, . . .} → {1, 2, . . .} there exists a uniformly continuous operator
Pr : K → F such that
lim Br(T ) · · · · · Br(1) x = Pr x
T →∞
for each x ∈ K.
(9.2)
(F,0)
(2) For each > 0 there exist a neighborhood U of {Bt }∞
t=1 in the space Aue
and a natural number N such that for each {Ct }∞
∈
U , each r : {1, 2 . . .} →
t=1
{1, 2, . . .} and each integer T ≥ N ,
Cr(T ) · · · · · Cr(1) x − Pr x ≤ ∀x ∈ K.
(9.3)
(F,0)
Proof. As in Section 8, given {At }∞
t=1 ∈ Aue , γ ∈ (0, 1), and an integer i ≥ 1, we
(F,0)
∞
define {Atγ }t=1 ∈ Aue (see (8.1)), a natural number N (γ , i) (see (8.5)) and an open
(F,0)
∞
(see property (a)). Again as in
neighborhood U ({At }∞
t=1 , γ , i) of {Atγ }t=1 in Aue
Section 8 we define a set F which is a countable intersection of open everywhere dense
(F,0)
sets in Aue by
∞
∞
(F,0)
F = ∩∞
q=1 ∪ U {At }t=1 , γ , i : {At }t=1 ∈ Aue , γ ∈ (0, 1), i = q, q + 1, . . . . (9.4)
S. Reich and A. J. Zaslavski
Assume that {Bt }∞
t=1 ∈ F and ∈ (0, 1). Choose a number 0 such that
80 (−1 rad(K) + 1 < 8−1 .
0 < 64−1 min{, (} ,
17
(9.5)
Choose a natural number q such that
64 · 2−q < 0 .
(F,0)
There exist {At }∞
t=1 ∈ Aue
, γ ∈ (0, 1), and an integer i ≥ q such that
∞
Bt t=1 ∈ U {At }∞
t=1 , γ , i .
(9.6)
(9.7)
It was shown in Section 8 (see (8.6)) that the following property holds:
(c) for each z ∈ K, each integer T ≥ N (γ , i), and each r : {1, . . . , T } → {1, 2, . . .},
d Ar(T )γ · · · · · Ar(1)γ z, F ≤ 8−i−1 .
(9.8)
By the definition of U ({At }∞
t=1 , γ , i) (see Section 8 and property (a)) the following
property holds:
∞
(d) for each {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i), each r : {1, . . . , N (γ , i)} → {1, 2, . . .}, and
each x ∈ K,
Cr(N(γ ,i)) · · · · · Cr(1) x − Ar(N (γ ,i))γ · · · · · Ar(1)γ x ≤ 16−1 · 8−i .
(9.9)
Assume that r : {1, 2, . . .} → {1, 2, . . .}. Then by property (c), for each x ∈ K there
exists fr (x) ∈ K such that
Ar(N (γ ,i))γ · · · · · Ar(1)γ x − fr (x) ≤ 2 · 8−i−1 .
(9.10)
∞
We show that for each {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i), each integer T ≥ N (γ , i), and each
x ∈ K,
Cr(T ) · · · · · Cr(1) x − fr (x) ≤ 8−1 .
(9.11)
∞
Let {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i) and let x ∈ K. By (9.10) and property (d),
Cr(N(γ ,i)) · · · · · Cr(1) x − fr (x) ≤ 8−i 16−1 + 4−1 .
Set
z = fr (x) + 8i ( Cr(N(γ ,i)) · · · · · Cr(1) x − fr (x) .
(9.12)
(9.13)
It follows from (9.12), (9.13), and the definition of ( that z ∈ K and
Cr(N(γ ,i)) · · · · · Cr(1) x = 8−i (−1 z + 1 − 8−i (−1 fr (x).
(9.14)
It follows from (9.14), (9.5), and (9.6) that for each integer T > N(γ , i),
Cr(T ) · · · · · Cr(1) x = 8−i (−1 Cr(T ) · · · · · Cr(N (γ ,i)+1) z + 1 − 8−i (−1 fr (x).
(9.15)
Together with (9.14) and (9.5) this implies that for each integer T ≥ N (γ , i),
Cr(T ) · · · · · Cr(1) x − fr (x) ≤ 2 rad(K)8−i (−1 < 8−1 .
(9.16)
18
Convergence of generic infinite products of affine operators
Therefore we have shown that for each r : {1, 2, . . .} → {1, 2, . . .} and each x ∈ K there
exists fr (x) ∈ F such that the following property holds:
∞
(e) for each {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i), each integer T ≥ N (γ , i), and each x ∈ K
the inequality (9.11) is valid.
Since is an arbitrary positive number this implies that for each r : {1, 2, . . .} →
{1, 2, . . .} there exists an operator Pr : K → K such that
lim Br(T ) · · · · · Br(1) x = Pr x,
T →∞
x ∈ K.
Let r : {1, 2, . . .} → {1, 2, . . .}. By (9.17), property (e), and (9.11),
Pr x − fr (x) ≤ 8−1 , x ∈ K,
(9.17)
(9.18)
∞
and for each {Ct }∞
t=1 ∈ U ({At }t=1 , γ , i), each integer T ≥ N (γ , i), and each x ∈ K,
Cr(T ) · · · · · Cr(1) x − Pr (x) ≤ 4−1 .
This completes the proof of Theorem 9.1.
(9.19)
Acknowledgments
The first author was partially supported by the Israel Science Foundation founded by
the Israel Academy of Sciences and Humanities, by the Fund for the Promotion of
Research at the Technion, and by the Technion VPR Fund.
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Simeon Reich: Department of Mathematics, The Technion-Israel Institute of Technology, 32000 Haifa, Israel
E-mail address: sreich@tx.technion.ac.il
Alexander J. Zaslavski: Department of Mathematics, The Technion-Israel Institute
of Technology, 32000 Haifa, Israel
E-mail address: ajzasl@tx.technion.ac.il